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Explicit dynamics
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framework/doc/content/source/timeintegrators/CentralDifference.md
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# CentralDifference | ||
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!syntax description /Executioner/TimeIntegrator/CentralDifference | ||
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## Description | ||
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Central difference integration is a common explicit integration scheme, typically used in structural dynamics. For a variable, $u$, at time $t$ with a time step, $\Delta t$, the central difference approximations for the first and second time derivatives, $\dot{u}$ and $\ddot{u}$, are given as, | ||
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\begin{equation} | ||
\begin{aligned} | ||
\mathbf{\dot{u}}(t) &=& \frac{\mathbf{u}(t+\Delta t)-\mathbf{u}(t-\Delta t)}{2 \Delta t} \\ | ||
\mathbf{\ddot{u}}(t) &=& \frac{\mathbf{u}(t+\Delta t)-2\mathbf{u}(t)+\mathbf{u}(t-\Delta t)}{\Delta t^2} | ||
\end{aligned} | ||
\end{equation} | ||
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The central difference time integrator derives from the [`ActuallyExplicitEuler`](ActuallyExplicitEuler.md) class and therefore circumvents the nonlinear solver. It can be used with `consistent`, `lumped` or `lump_preconditioned`, `solve_type` options. Information on these solve options can be found on the [`ActuallyExplicitEuler`](ActuallyExplicitEuler.md) page. | ||
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!syntax parameters /Executioner/TimeIntegrator/CentralDifference | ||
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!syntax inputs /Executioner/TimeIntegrator/CentralDifference | ||
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!syntax children /Executioner/TimeIntegrator/CentralDifference |
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